IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 48, NO. 1, 2000

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Computationally Efficient Optimal Power Allocation Algorithms for Multicarrier Communication Systems Brian S. Krongold, Kannan Ramchandran, Member, IEEE, and Douglas L. Jones, Senior Member, IEEE

Abstract—In this paper, we present an optimal, computationally efficient, integer-bit power allocation algorithm for discrete multitone modulation. Using efficient lookup table searches and a Lagrange-multiplier bisection search, our algorithm converges faster to the optimal solution than existing techniques and can replace the use of suboptimal methods because of its low computational complexity. Fast algorithms are developed for the data rate and performance margin maximization problems. Index Terms—Discrete multitone modulation, loading algorithm, multicarrier communication systems, power allocation.

II. OPTIMAL POWER ALLOCATION PROBLEM Maximizing channel capacity in a spectrally-shaped Gaussian channel is achieved by the well-known waterpouring distribution [9]. However, this distribution is not well suited for practical data transmission because it assumes noninteger-bit constellations, does not obey a given probability of error, and is difficult to compute. Instead, the data throughput optimization problem [3] is of more practical importance subject to

I. INTRODUCTION

R

ESEARCH in multicarrier modulation has grown tremendously in recent years due to the demand for high-speed data transmission over twisted-pair copper wiring, an environment where severe intersymbol interference (ISI) can occur [1], [2]. Instead of employing single-carrier modulation with a very complex adaptive equalizer, the channel is divided into subchannels that are essentially ISI-free independent additive white is sufficiently Gaussian noise (AWGN) channels, provided large. Although multicarrier modulation eliminates the need for an expensive equalizer, it creates a new problem: given some power budget, how should power and bits be allocated to each subchannel in order to maximize performance? Many algorithms for allocating power among subchannels exist; however, these methods are either suboptimal and computationally efficient [3]–[6] or optimal but slow to obtain the power allocation [7]. In this paper, we present practical and efficient discrete multitone modulation (DMT) loading algorithms that are guaranteed to converge to the optimal power allocation solution. The algorithms use efficient lookup tables and a fast Lagrange bisection search that is popular in the image compression community [8].

Paper approved by Y. Li, the Editor for Wireless Communications Theory of the IEEE Communications Society. Manuscript received March 18, 1998; revised December 30, 1998 and July 17, 1999. This work was supported in part by the Office of Naval Research under the Young Investigator Award N00014-97-1-0864 and Contract N00014-95-1-0674, and the Joint Services Electronics Program (JSEP) under Award N00014-96-1-0129. This paper was presented in part at ICC ’98, Atlanta, GA, June 1998. B. S. Krongold and D. L. Jones are with the Department of Electrical and Computer Engineering and the Coordinated Science Laboratory, University of Illinois at Urbana-Champaign, Urbana, IL 61801 USA (e-mail: [email protected]; [email protected]). K. Ramchandran was with the Department of Electrical and Computer Engineering, University of Illinois at Urbana-Champaign, Urbana, IL 61801 USA. He is now with the Department of Electrical Engineering and Computer Science, University of California at Berkeley, Berkeley, CA 94720 USA (e-mail: [email protected]). Publisher Item Identifier S 0090-6778(00)00497-9.

and

(2.1)

are the rate (in bits/symbol), allocated where , , and power, and error probability, respectively, of the th subchannel, is a fixed error-probability constraint, and is a total power constraint. An additional constraint (of significant to be an integer number practical importance) is restricting of bits/symbol. We will enforce this condition later and assume can be any nonnegative real number. for now that A. Lagrange Solution The optimization problem in (2.1) can by reformulated as an unconstrainted optimization problem1 by merging rate and power through the Lagrange multiplier (2.2) is the Lagrange cost and . Each minimum where Lagrange cost for a fixed corresponds to the optimal power allocation for some total power budget. For a fixed , the Lagrange cost is minimized when , for all , or more precisely for

(2.3)

is a function of that satisfies the error-probability where constraint with equality.2 Thus, the cost is minimized when the rates and powers for each subchannel are chosen to correspond to the point on the rate-versus-power curve with slope . The total power allocated for a fixed is obtained by simply summing the power allocated to the subchannels. The goal is to find 1This reformulation is equivalent provided that rate is a convex function of power, which is the case in virtually every standard class of signal constellations, including quadrature amplitude modulation (QAM). When this is not true, the solution is only optimal to within a convex-hull approximation. 2It can be shown that meeting the error-probability constraint with equality is optimal.

0090–6778/00$10.00 © 2000 IEEE

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IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 48, NO. 1, 2000

the optimal such that the total power allocated equals the in the problem. given value of An additional formulation can be derived by defining the signal-to-noise ratio (SNR) of the th subchannel to be and the channel-to-noise ratio (CNR) to be , where is the symbol period, is the subchannel power gain, and is the one-dimensional subchannel noise power. Applying the chain rule to (2.3), the following cost minimization criterion is obtained, which will be used to develop efficient loading algorithms proposed later (2.4) (a)

B. Integer-Bit Restriction Enforcing the restriction of integer-bit constellations, we obtain a sampled version of the continuous rate-power curves at operating points, which constitute the only admissible rate-power combinations. The optimal operating point for the th subchannel for a given can be shown to be the point which is first “impinged upon” by a “plane-wave” of slope , as shown in Fig. 1(a) [8]. Rather than a unique value of , the discrete nature of the problem results in each operating point having a continuous range of optimal values associated with it as shown in Fig. 1(b). The combination of all possible subchannel rate-power combinations summed together gives the composite rate-power function, and an example of this is shown in Fig. 2. The upper-leftmost operating points are the set of all possible optimal operating points, and the lines connecting them form , the convex hull of the composite function. For a given the optimal operating point is the one on the convex hull with power closest to, without exceeding, the power budget. Furthermore, the Lagrange solution will always obtain the convex hull solution and, hence, the optimal operating point.

(b) Fig. 1. (a) Depiction of a plane wave of slope impinging upon the rate-power convex hull.(b) Illustration of nonunique slopes for rate-power operating points.

III. FAST ALGORITHM FOR POWER ALLOCATION We now develop the fast integer-bit loading algorithm for data-rate maximization. Because computing all the composite rate-power operating points is much too expensive, a more efficient approach is to iteratively search for a . This can be done by evaluating a chosen for its corresponding total power, followed by an update to get closer to an optimal solution. A. Fast Power Allocation via Table Lookup Each encountered during the search must be evaluated to determine the total power associated with it and requires computing the optimal operating point for each subchannel on the rate-power function using (2.3), summing the power allocated . to the subchannels and comparing the result to Direct computation of the optimal operating point for each subchannel can be avoided by using the slope nonuniqueness property, shown in Fig. 1(b), and precomputing lookup tables of operating point slope bounds. Evaluating a given can easily be done for each subchannel by finding which slope range falls in and assigning the corresponding rate and power. The lookup

Fig. 2. Composite rate-power curve for three subchannels with up to 5 bits/symbol. It can be seen that more than one point can operate at the maximum rate with a total power less than P . However, only one point with the maximum rate can be on the convex hull, and it is always the most power-efficient solution.

tables can be generated from the rate-SNR characteristics of the channel, which are invariant to the channel conditions.

IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 48, NO. 1, 2000

25

For a given , the Lagrange minimization formulation of (2.4) states that the optimal operating point is found using the slope of the rate-SNR function of each subchannel. Due to the discrete number of operating points, rate-SNR ranges can also be precomputed and placed into a lookup table to avoid real-time computation. In practice, all or most of the subchannels have identical rate-SNR operating characteristics (i.e., constraint are the the available signal constellations and ’s. Consame), and the difference between them are the sequently, the ranges for these channels will be identical, and only one lookup table need be stored,3 resulting in a significant memory reduction. Following the computation of for a subchannel and using the lookup table to find the rate-SNR operating point, the allocated power and rates are computed as

Fig. 3. Illustration of bisection method slope update.

(3.1)

B. Bisection Method for Fast

decibels) that a system can tolerate while still operating under the bit-error-probability constraint [4]. The performance margin optimization problem for a given target rate is as follows:

Convergence

There are two major problems that can be encountered when trying to find a : fast convergence and the ability to recognize has been reached. A bisection method (similar to when a the false position method [10]) solves both of these problems and exploits the monotonic relationship between and through a binary search-like procedure [8]. The bisection method uses two previously evaluated slope and corresponding to total powers and values (which are below and above , respectively) and and . The bisection method simply lowers total rates and by computing the following the gap between updated slope on the composite rate-power curve: (3.2) is then evaluated. If The total power corresponding to is greater than , we update with while the same. The opposite update is done if is keeping . An example of the bisection method slope upless than date is shown in Fig. 3. The slope update procedure is repeated equals either or , and the power allocation until is chosen to load the multicarrier system.4 corresponding to C. Performance Margin Optimization Another important quantity of interest in DMT systems is the , which is the amount of noise (in performance margin5 3For any remaining subchannels with different rate-SNR characteristics, different lookup tables need to be defined. 4It is highly improbable that P will exactly equal P , but if this does occur, the algorithm allocates power according to . 5For example, if a single channel requires a 12-dB SNR to operate at some data rate with P , then providing 18 db of SNR results in a +6-dB performance margin.

subject to

and (3.3) Since performance margin is simply a scaling of the zero-margin allocated power in each subchannel by a constant amount, the minimum power allocation needed to meet the rate target with zero margin can be found, and the resulting powers can be scaled to utilize the total power budget. This new optimization problem is as follows:

subject to

and (3.4) In this case, only the convex hull operating points of the composite rate-power function can be optimal solutions because there is no leftover power as in the rate maximization problem. An algorithm very similar to the rate maximization algorithm can be used to solve (3.4), with the difference that is up[11]. Once this dated so that the total rate converges to has been done, the final power allocation is computed as , where .

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IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 48, NO. 1, 2000

(a)

(b)

(c)

(d)

Fig. 4. (a) SNR (in decibels) for 256 subchannels. (b) Optimal power allocation. (c) Optimal bit allocation.(d) Total bits allocated versus bisection iteration number.

TABLE I ALGORITHM PSEUDOCODE

D. Algorithm Implementation Pseudocode for the rate maximization algorithm using rate-SNR lookup tables is listed in Table I. Initial loading and , which are of the system may require initial far from the optimum values to ensure that the condition in step 1 is met. Simple worst case initializations are the point

with zero rate and power and the point with maximum rate and power ( ). Prior knowledge of typical channels can help the designer choose initial values closer to the range of optimal values, which will allow the algorithm to converge takes faster. Convergence of the bisection method to a iterations. Assuming is approximately additions, precomputed, each iteration requires at most multiplies, and lookup table evaluations. 1 division, The resulting computational complexity of the algorithm is . Knowledge of previous iteration lookup results can be used to drastically reduce complexity [12] as some subchannels converge rather quickly, and most subchannel rate-power assignments do not change in the final one-third or so iterations. Fig. 4 shows an example of our algorithm for a test channel where the available signal constellations were 0–10 bits/symbol QAM and a symbol error probability constraint of 10 was imposed on each subchannel. The algorithm converged very quickly6 to the optimal solution with 14 bisection search iterations using very conservative initial low and high slope values. 6After

only eight iterations, 98.8% of the optimal rate is achieved.

IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 48, NO. 1, 2000

For tracking scenarios, when the channel conditions change only slightly, an optimal value may not be very different from the previous optimal one. Therefore, initial low and high values can be chosen much closer to obtain faster convergence.

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TABLE II COMPARISON OF MARGIN MAXIMIZATION ALGORITHMS

IV. COMPARISONS AND CONCLUSIONS The Hughes–Hartog algorithm [7] is an optimal loading algorithm which achieves the solution by adding one bit at a time to the channel requiring the smallest additional power to increase its rate. Whereas this technique can be used to solve both data rate and margin maximization, the algorithm requires an intensive amount of sorting and converges very slowly in practical DMT scenarios [3]. Our algorithms achieve exactly the same solutions and are cheaper to implement. The algorithm in [4] attempts to maximize margin in a suboptimal fashion that relies on rounding to integer rates. Another disadvantage of this algorithm is its use of the SNR gap approximation [2] to allocate bits to its subchannels. Furthermore, in the final part of the algorithm, it requires a modest amount of sorting to subtract or add bits one at a time to meet the target bit rate. The overall complexity is less than Hughes–Hartog, but is approximately the same or slightly more than the proposed margin algorithm in this paper. The algorithm in [6] attempts to maximize the subchannel SNR’s rather than the margin and again relies on rounding. Whereas this is a different criterion for loading, the resulting allocation should be extremely close if not identical. The results in [6] in show improvement of overall SNR compared to [4] as well as some reduction in complexity. As in [4], it uses a modest amount of sorting to subtract or add bits one at a time, which may be expensive if the initial part of the algorithm is too far from the target rate. The overall complexity of the algorithm is dominated by searches and additions, but the operation count will typically be on the same order as the margin algorithm in this paper. Using the same channel SNR’s shown in Fig. 4(a), optimal margin maximization is compared to the two algorithms described above. In the case of [6], the final rate assignment is used and power is allocated to meet the 10 symbol-error-probability constraint. The target data rate is 500 bits/symbol which corresponds to 2 Mbp/s for a multicarrier symbol rate of 4 kHz, and QAM signal constellations with a range of 2–10 bits/symbol are employed.7 Table II shows the resulting margins for the example channel and with AWGN levels of +3, −3, and −6 dB from the original. As can be seen, the method of [6] is very close to the optimal margin, with about a 2% margin differential for all conditions. However, the method of [4] has significant performance variation with different amounts of AWGN added to the channel. This is due to the SNR gap becoming even more suboptimal as subchannel SNR’s decrease and the number of assigned bits in a subchannel also decreases.

The rate maximization algorithms in [3] and [5] are also based upon the SNR gap approximation combined with rounding, and they suffer from its drawbacks as well. Both methods require moderate amounts of sorting, in addition to mathematical computation, and the resulting complexities are probably higher than the proposed optimal algorithm. Although we developed our algorithms specifically for integer-bit constellations, they can be used for systems containing noninteger-bit constellations as well. In fact, the algorithms will work for any set of discrete points on the rate-SNR curve provided the function is convex, which it almost always will be. Thus, these algorithms should be considered optimal loading algorithms for any discrete set of available signal constellations. REFERENCES [1] J. A. C. Bingham, “Multicarrier modulation for data transmission: An idea whose time has come,” IEEE Commun. Mag., vol. 28, pp. 5–14, May 1990. [2] J. M. Cioffi, “A multicarrier primer,” ANSI T1E1.4 Committee Contribution, Nov. 1991. [3] P. S. Chow, “Bandwidth optimized digital transmission techniques for spectrally shaped channels with impulsive noise,” Ph.D. dissertation, Stanford Univ., Stanford, CA, 1993. [4] P. S. Chow, J. M. Cioffi, and J. A. C. Bingham, “A practical discrete multitone transceiver loading algorithm for data transmission over spectrally shaped channels,” IEEE Trans. Commun., vol. 43, pp. 773–775, Feb./Mar./Apr. 1995. [5] A. Leke and J. M. Cioffi, “A maximum rate loading algorithm for discrete multitone modulation systems,” in Proc. IEEE GLOBECOM’97, Phoenix, AZ, Nov. 1997, pp. 1514–1518. [6] R. F. H. Fischer and J. B. Huber, “A new loading algorithm for discrete multitone transmission,” in Proc. IEEE GLOBECOM’96, London, U.K., Nov. 1996, pp. 724–728. [7] D. Hughes-Hartogs, “Ensemble modem structure for imperfect transmission media,” 4 679 227 (July 1987), 4 731 816 (March 1988) and 4 833 796 (May 1989). [8] K. Ramchandran and M. Vetterli, “Best wavelet packet bases in a ratedistortion sense,” IEEE Trans. Acoust., Speech, Signal Processing, vol. 2, pp. 160–175, Apr. 1993. [9] R. G. Gallager, Information Theory and Realiable Communication. New York: Wiley, 1968. [10] D. G. Luenberger, Introduction to Linear and Nonlinear Programming, 2nd ed. Reading, MA: Addison-Wesley, 1989. [11] B. S. Krongold, “Power and bandwidth optimization for multicarrier communication systems,” M.S. thesis, Univ. Illinois at Urbana–Champaign, 1997. [12] B. S. Krongold, K. Ramchandran, and D. L. Jones, “Section division operating point slope determination method for multicarrier communication systems,” pending. 7Both the proposed algorithm and [6] can achieve slight performance gains by allowing 1 bit/symbol as a signaling choice. However, [4] often yields poorer solutions in this case.

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Computationally Efficient Optimal Power Allocation Algorithms for Multicarrier Communication Systems Brian S. Krongold, Kannan Ramchandran, Member, IEEE, and Douglas L. Jones, Senior Member, IEEE

Abstract—In this paper, we present an optimal, computationally efficient, integer-bit power allocation algorithm for discrete multitone modulation. Using efficient lookup table searches and a Lagrange-multiplier bisection search, our algorithm converges faster to the optimal solution than existing techniques and can replace the use of suboptimal methods because of its low computational complexity. Fast algorithms are developed for the data rate and performance margin maximization problems. Index Terms—Discrete multitone modulation, loading algorithm, multicarrier communication systems, power allocation.

II. OPTIMAL POWER ALLOCATION PROBLEM Maximizing channel capacity in a spectrally-shaped Gaussian channel is achieved by the well-known waterpouring distribution [9]. However, this distribution is not well suited for practical data transmission because it assumes noninteger-bit constellations, does not obey a given probability of error, and is difficult to compute. Instead, the data throughput optimization problem [3] is of more practical importance subject to

I. INTRODUCTION

R

ESEARCH in multicarrier modulation has grown tremendously in recent years due to the demand for high-speed data transmission over twisted-pair copper wiring, an environment where severe intersymbol interference (ISI) can occur [1], [2]. Instead of employing single-carrier modulation with a very complex adaptive equalizer, the channel is divided into subchannels that are essentially ISI-free independent additive white is sufficiently Gaussian noise (AWGN) channels, provided large. Although multicarrier modulation eliminates the need for an expensive equalizer, it creates a new problem: given some power budget, how should power and bits be allocated to each subchannel in order to maximize performance? Many algorithms for allocating power among subchannels exist; however, these methods are either suboptimal and computationally efficient [3]–[6] or optimal but slow to obtain the power allocation [7]. In this paper, we present practical and efficient discrete multitone modulation (DMT) loading algorithms that are guaranteed to converge to the optimal power allocation solution. The algorithms use efficient lookup tables and a fast Lagrange bisection search that is popular in the image compression community [8].

Paper approved by Y. Li, the Editor for Wireless Communications Theory of the IEEE Communications Society. Manuscript received March 18, 1998; revised December 30, 1998 and July 17, 1999. This work was supported in part by the Office of Naval Research under the Young Investigator Award N00014-97-1-0864 and Contract N00014-95-1-0674, and the Joint Services Electronics Program (JSEP) under Award N00014-96-1-0129. This paper was presented in part at ICC ’98, Atlanta, GA, June 1998. B. S. Krongold and D. L. Jones are with the Department of Electrical and Computer Engineering and the Coordinated Science Laboratory, University of Illinois at Urbana-Champaign, Urbana, IL 61801 USA (e-mail: [email protected]; [email protected]). K. Ramchandran was with the Department of Electrical and Computer Engineering, University of Illinois at Urbana-Champaign, Urbana, IL 61801 USA. He is now with the Department of Electrical Engineering and Computer Science, University of California at Berkeley, Berkeley, CA 94720 USA (e-mail: [email protected]). Publisher Item Identifier S 0090-6778(00)00497-9.

and

(2.1)

are the rate (in bits/symbol), allocated where , , and power, and error probability, respectively, of the th subchannel, is a fixed error-probability constraint, and is a total power constraint. An additional constraint (of significant to be an integer number practical importance) is restricting of bits/symbol. We will enforce this condition later and assume can be any nonnegative real number. for now that A. Lagrange Solution The optimization problem in (2.1) can by reformulated as an unconstrainted optimization problem1 by merging rate and power through the Lagrange multiplier (2.2) is the Lagrange cost and . Each minimum where Lagrange cost for a fixed corresponds to the optimal power allocation for some total power budget. For a fixed , the Lagrange cost is minimized when , for all , or more precisely for

(2.3)

is a function of that satisfies the error-probability where constraint with equality.2 Thus, the cost is minimized when the rates and powers for each subchannel are chosen to correspond to the point on the rate-versus-power curve with slope . The total power allocated for a fixed is obtained by simply summing the power allocated to the subchannels. The goal is to find 1This reformulation is equivalent provided that rate is a convex function of power, which is the case in virtually every standard class of signal constellations, including quadrature amplitude modulation (QAM). When this is not true, the solution is only optimal to within a convex-hull approximation. 2It can be shown that meeting the error-probability constraint with equality is optimal.

0090–6778/00$10.00 © 2000 IEEE

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IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 48, NO. 1, 2000

the optimal such that the total power allocated equals the in the problem. given value of An additional formulation can be derived by defining the signal-to-noise ratio (SNR) of the th subchannel to be and the channel-to-noise ratio (CNR) to be , where is the symbol period, is the subchannel power gain, and is the one-dimensional subchannel noise power. Applying the chain rule to (2.3), the following cost minimization criterion is obtained, which will be used to develop efficient loading algorithms proposed later (2.4) (a)

B. Integer-Bit Restriction Enforcing the restriction of integer-bit constellations, we obtain a sampled version of the continuous rate-power curves at operating points, which constitute the only admissible rate-power combinations. The optimal operating point for the th subchannel for a given can be shown to be the point which is first “impinged upon” by a “plane-wave” of slope , as shown in Fig. 1(a) [8]. Rather than a unique value of , the discrete nature of the problem results in each operating point having a continuous range of optimal values associated with it as shown in Fig. 1(b). The combination of all possible subchannel rate-power combinations summed together gives the composite rate-power function, and an example of this is shown in Fig. 2. The upper-leftmost operating points are the set of all possible optimal operating points, and the lines connecting them form , the convex hull of the composite function. For a given the optimal operating point is the one on the convex hull with power closest to, without exceeding, the power budget. Furthermore, the Lagrange solution will always obtain the convex hull solution and, hence, the optimal operating point.

(b) Fig. 1. (a) Depiction of a plane wave of slope impinging upon the rate-power convex hull.(b) Illustration of nonunique slopes for rate-power operating points.

III. FAST ALGORITHM FOR POWER ALLOCATION We now develop the fast integer-bit loading algorithm for data-rate maximization. Because computing all the composite rate-power operating points is much too expensive, a more efficient approach is to iteratively search for a . This can be done by evaluating a chosen for its corresponding total power, followed by an update to get closer to an optimal solution. A. Fast Power Allocation via Table Lookup Each encountered during the search must be evaluated to determine the total power associated with it and requires computing the optimal operating point for each subchannel on the rate-power function using (2.3), summing the power allocated . to the subchannels and comparing the result to Direct computation of the optimal operating point for each subchannel can be avoided by using the slope nonuniqueness property, shown in Fig. 1(b), and precomputing lookup tables of operating point slope bounds. Evaluating a given can easily be done for each subchannel by finding which slope range falls in and assigning the corresponding rate and power. The lookup

Fig. 2. Composite rate-power curve for three subchannels with up to 5 bits/symbol. It can be seen that more than one point can operate at the maximum rate with a total power less than P . However, only one point with the maximum rate can be on the convex hull, and it is always the most power-efficient solution.

tables can be generated from the rate-SNR characteristics of the channel, which are invariant to the channel conditions.

IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 48, NO. 1, 2000

25

For a given , the Lagrange minimization formulation of (2.4) states that the optimal operating point is found using the slope of the rate-SNR function of each subchannel. Due to the discrete number of operating points, rate-SNR ranges can also be precomputed and placed into a lookup table to avoid real-time computation. In practice, all or most of the subchannels have identical rate-SNR operating characteristics (i.e., constraint are the the available signal constellations and ’s. Consame), and the difference between them are the sequently, the ranges for these channels will be identical, and only one lookup table need be stored,3 resulting in a significant memory reduction. Following the computation of for a subchannel and using the lookup table to find the rate-SNR operating point, the allocated power and rates are computed as

Fig. 3. Illustration of bisection method slope update.

(3.1)

B. Bisection Method for Fast

decibels) that a system can tolerate while still operating under the bit-error-probability constraint [4]. The performance margin optimization problem for a given target rate is as follows:

Convergence

There are two major problems that can be encountered when trying to find a : fast convergence and the ability to recognize has been reached. A bisection method (similar to when a the false position method [10]) solves both of these problems and exploits the monotonic relationship between and through a binary search-like procedure [8]. The bisection method uses two previously evaluated slope and corresponding to total powers and values (which are below and above , respectively) and and . The bisection method simply lowers total rates and by computing the following the gap between updated slope on the composite rate-power curve: (3.2) is then evaluated. If The total power corresponding to is greater than , we update with while the same. The opposite update is done if is keeping . An example of the bisection method slope upless than date is shown in Fig. 3. The slope update procedure is repeated equals either or , and the power allocation until is chosen to load the multicarrier system.4 corresponding to C. Performance Margin Optimization Another important quantity of interest in DMT systems is the , which is the amount of noise (in performance margin5 3For any remaining subchannels with different rate-SNR characteristics, different lookup tables need to be defined. 4It is highly improbable that P will exactly equal P , but if this does occur, the algorithm allocates power according to . 5For example, if a single channel requires a 12-dB SNR to operate at some data rate with P , then providing 18 db of SNR results in a +6-dB performance margin.

subject to

and (3.3) Since performance margin is simply a scaling of the zero-margin allocated power in each subchannel by a constant amount, the minimum power allocation needed to meet the rate target with zero margin can be found, and the resulting powers can be scaled to utilize the total power budget. This new optimization problem is as follows:

subject to

and (3.4) In this case, only the convex hull operating points of the composite rate-power function can be optimal solutions because there is no leftover power as in the rate maximization problem. An algorithm very similar to the rate maximization algorithm can be used to solve (3.4), with the difference that is up[11]. Once this dated so that the total rate converges to has been done, the final power allocation is computed as , where .

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IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 48, NO. 1, 2000

(a)

(b)

(c)

(d)

Fig. 4. (a) SNR (in decibels) for 256 subchannels. (b) Optimal power allocation. (c) Optimal bit allocation.(d) Total bits allocated versus bisection iteration number.

TABLE I ALGORITHM PSEUDOCODE

D. Algorithm Implementation Pseudocode for the rate maximization algorithm using rate-SNR lookup tables is listed in Table I. Initial loading and , which are of the system may require initial far from the optimum values to ensure that the condition in step 1 is met. Simple worst case initializations are the point

with zero rate and power and the point with maximum rate and power ( ). Prior knowledge of typical channels can help the designer choose initial values closer to the range of optimal values, which will allow the algorithm to converge takes faster. Convergence of the bisection method to a iterations. Assuming is approximately additions, precomputed, each iteration requires at most multiplies, and lookup table evaluations. 1 division, The resulting computational complexity of the algorithm is . Knowledge of previous iteration lookup results can be used to drastically reduce complexity [12] as some subchannels converge rather quickly, and most subchannel rate-power assignments do not change in the final one-third or so iterations. Fig. 4 shows an example of our algorithm for a test channel where the available signal constellations were 0–10 bits/symbol QAM and a symbol error probability constraint of 10 was imposed on each subchannel. The algorithm converged very quickly6 to the optimal solution with 14 bisection search iterations using very conservative initial low and high slope values. 6After

only eight iterations, 98.8% of the optimal rate is achieved.

IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 48, NO. 1, 2000

For tracking scenarios, when the channel conditions change only slightly, an optimal value may not be very different from the previous optimal one. Therefore, initial low and high values can be chosen much closer to obtain faster convergence.

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TABLE II COMPARISON OF MARGIN MAXIMIZATION ALGORITHMS

IV. COMPARISONS AND CONCLUSIONS The Hughes–Hartog algorithm [7] is an optimal loading algorithm which achieves the solution by adding one bit at a time to the channel requiring the smallest additional power to increase its rate. Whereas this technique can be used to solve both data rate and margin maximization, the algorithm requires an intensive amount of sorting and converges very slowly in practical DMT scenarios [3]. Our algorithms achieve exactly the same solutions and are cheaper to implement. The algorithm in [4] attempts to maximize margin in a suboptimal fashion that relies on rounding to integer rates. Another disadvantage of this algorithm is its use of the SNR gap approximation [2] to allocate bits to its subchannels. Furthermore, in the final part of the algorithm, it requires a modest amount of sorting to subtract or add bits one at a time to meet the target bit rate. The overall complexity is less than Hughes–Hartog, but is approximately the same or slightly more than the proposed margin algorithm in this paper. The algorithm in [6] attempts to maximize the subchannel SNR’s rather than the margin and again relies on rounding. Whereas this is a different criterion for loading, the resulting allocation should be extremely close if not identical. The results in [6] in show improvement of overall SNR compared to [4] as well as some reduction in complexity. As in [4], it uses a modest amount of sorting to subtract or add bits one at a time, which may be expensive if the initial part of the algorithm is too far from the target rate. The overall complexity of the algorithm is dominated by searches and additions, but the operation count will typically be on the same order as the margin algorithm in this paper. Using the same channel SNR’s shown in Fig. 4(a), optimal margin maximization is compared to the two algorithms described above. In the case of [6], the final rate assignment is used and power is allocated to meet the 10 symbol-error-probability constraint. The target data rate is 500 bits/symbol which corresponds to 2 Mbp/s for a multicarrier symbol rate of 4 kHz, and QAM signal constellations with a range of 2–10 bits/symbol are employed.7 Table II shows the resulting margins for the example channel and with AWGN levels of +3, −3, and −6 dB from the original. As can be seen, the method of [6] is very close to the optimal margin, with about a 2% margin differential for all conditions. However, the method of [4] has significant performance variation with different amounts of AWGN added to the channel. This is due to the SNR gap becoming even more suboptimal as subchannel SNR’s decrease and the number of assigned bits in a subchannel also decreases.

The rate maximization algorithms in [3] and [5] are also based upon the SNR gap approximation combined with rounding, and they suffer from its drawbacks as well. Both methods require moderate amounts of sorting, in addition to mathematical computation, and the resulting complexities are probably higher than the proposed optimal algorithm. Although we developed our algorithms specifically for integer-bit constellations, they can be used for systems containing noninteger-bit constellations as well. In fact, the algorithms will work for any set of discrete points on the rate-SNR curve provided the function is convex, which it almost always will be. Thus, these algorithms should be considered optimal loading algorithms for any discrete set of available signal constellations. REFERENCES [1] J. A. C. Bingham, “Multicarrier modulation for data transmission: An idea whose time has come,” IEEE Commun. Mag., vol. 28, pp. 5–14, May 1990. [2] J. M. Cioffi, “A multicarrier primer,” ANSI T1E1.4 Committee Contribution, Nov. 1991. [3] P. S. Chow, “Bandwidth optimized digital transmission techniques for spectrally shaped channels with impulsive noise,” Ph.D. dissertation, Stanford Univ., Stanford, CA, 1993. [4] P. S. Chow, J. M. Cioffi, and J. A. C. Bingham, “A practical discrete multitone transceiver loading algorithm for data transmission over spectrally shaped channels,” IEEE Trans. Commun., vol. 43, pp. 773–775, Feb./Mar./Apr. 1995. [5] A. Leke and J. M. Cioffi, “A maximum rate loading algorithm for discrete multitone modulation systems,” in Proc. IEEE GLOBECOM’97, Phoenix, AZ, Nov. 1997, pp. 1514–1518. [6] R. F. H. Fischer and J. B. Huber, “A new loading algorithm for discrete multitone transmission,” in Proc. IEEE GLOBECOM’96, London, U.K., Nov. 1996, pp. 724–728. [7] D. Hughes-Hartogs, “Ensemble modem structure for imperfect transmission media,” 4 679 227 (July 1987), 4 731 816 (March 1988) and 4 833 796 (May 1989). [8] K. Ramchandran and M. Vetterli, “Best wavelet packet bases in a ratedistortion sense,” IEEE Trans. Acoust., Speech, Signal Processing, vol. 2, pp. 160–175, Apr. 1993. [9] R. G. Gallager, Information Theory and Realiable Communication. New York: Wiley, 1968. [10] D. G. Luenberger, Introduction to Linear and Nonlinear Programming, 2nd ed. Reading, MA: Addison-Wesley, 1989. [11] B. S. Krongold, “Power and bandwidth optimization for multicarrier communication systems,” M.S. thesis, Univ. Illinois at Urbana–Champaign, 1997. [12] B. S. Krongold, K. Ramchandran, and D. L. Jones, “Section division operating point slope determination method for multicarrier communication systems,” pending. 7Both the proposed algorithm and [6] can achieve slight performance gains by allowing 1 bit/symbol as a signaling choice. However, [4] often yields poorer solutions in this case.